Consider all triangles which are on the plane of the axis of a cone, have the apex of the cone as one of their vertices, and the diameter of the base as their bases. Do they have equal areas, are somehow related, or share any interesting properties? If so, could you show me the proof?

Consider all triangles which are on the plane of the axis of a cone,
have the apex of the cone as one of their vertices,
and the diameter of the base as their bases.
Do they have equal areas, are somehow related, or share any interesting properties?
If so, could you show me the proof?